Question

How do you express $132$ degrees in radians?

Answer 1

Hint: We are given with an angle in degrees which we have to express in radians. We know that, \[{{180}^{\circ }}=\pi \text{ radians}\] as \[2\pi \] would mean a complete circle which is \[{{360}^{\circ }}\]. So, \[{{1}^{\circ }}=\dfrac{\pi }{{{180}^{\circ }}}\text{ radians}\], we will multiply the given angle in degrees by \[\dfrac{\pi }{{{180}^{\circ }}}\] to express the angle in radians. Reducing it further, cancelling the common terms, we will have the angle in radians.

Complete step-by-step solution:
According to the given question, we have been given an angle which is expressed in degrees, that is, we have $132$ degrees. We have to now express this angle in radians.
We will begin with writing the given angle in degrees, we have,
$132$ degrees -----(1)
We know that a complete circle measures \[{{360}^{\circ }}\] which in radian terms would be \[2\pi \]. So, for a half circle that is \[{{180}^{\circ }}\] we have the angle in radian terms as \[\pi \text{ radians}\].
We can now write it as:
\[{{360}^{\circ }}=2\pi \text{ radians}\]
\[{{180}^{\circ }}=\pi \text{ radians}\]
So, for 1 degree we have,
\[{{1}^{\circ }}=\dfrac{\pi }{{{180}^{\circ }}}\text{ radians}\]
We will use this conversion factor to get the angle in the required unit, that is, we will multiply the given angle in degrees by \[\dfrac{\pi }{{{180}^{\circ }}}\] and further solving which gives us the angle in radians.
We have,
If \[{{1}^{\circ }}\to \dfrac{\pi }{{{180}^{\circ }}}\text{ radians}\]
Then for $132$ degrees,
\[\Rightarrow {{132}^{\circ }}\to \dfrac{\pi }{{{180}^{\circ }}}\times {{132}^{\circ }}\text{ radians}\]
We will now solve \[\dfrac{\pi }{{{180}^{\circ }}}\times {{132}^{\circ }}\text{ radians}\] and we get,
\[\Rightarrow \dfrac{\pi }{{{180}^{\circ }}}\times {{132}^{\circ }}\text{ radians}\]
We see that, numerator and the denominator have a multiple of 12. So, we will divide and multiply by 12 and reduce the expression, we have,
\[\Rightarrow \dfrac{\pi }{{{180}^{\circ }}}\times {{132}^{\circ }}\times \dfrac{12}{12}\]
Reducing the terms, we get,
\[\Rightarrow \dfrac{\pi }{15}\times 11\]
\[\Rightarrow \dfrac{11\pi }{15}\]
Therefore, we get, \[{{132}^{\circ }}=\dfrac{11\pi }{15}\text{ radians}\].

Note: The conversion factor for degrees to radians is \[{{1}^{\circ }}=\dfrac{\pi }{{{180}^{\circ }}}\text{ radians}\].
Similarly, the conversion factor for radians to degrees is \[1 \text{ radians}=\dfrac{{{180}^{\circ }}}{\pi }\deg .\]
The conversion factor should be carefully written and calculation should be done in a proper sequence to avoid errors.